J. Aust. Math. Soc.  75 (2003), 23-40
On products in lattice-ordered algebras

Karim Boulabiar
  Département des Classes Préparatoires
  Institut Préparatoire aux Etudes Scientifiques et Techniques
  Université 7 Novembre à Carthage
  BP 51, 2070-La Marsa
  Tunisia
  karim.boulabiar@ipest.rnu.tn


Abstract
Let $A$ be a uniformly complete vector sublattice of an Archimedean semiprime $f$-algebra $B$ and $p\in\{1,2,\dots\}$. It is shown that the set $\Pi_p^{B}(A) =\{f_1\cdots f_p:f_k\in A, k=1,\dots,p\}$ is a uniformly complete vector sublattice of $B$. Moreover, if $A$ is provided with an almost $f$-algebra multiplication $\ast$ then there exists a positive operator $T_p$ from $\Pi_p^{B}(A)$ into $A$ such that $f_1\ast\cdots\ast f_p=T_p(f_1\cdots f_p)$ for all $f_1,\dots,f_p\in A$. As application, being given a uniformly complete almost $f$-algebra $(A,\ast) $ and a natural number $p\geq3$, the set $\Pi_p^{\ast}(A) =\{ f_1\ast\cdots\ast f_p:f_k\in A, k=1,\dots,p\}$ is a uniformly complete semiprime $f$-algebra under the ordering and the multiplication inherited from $A$.
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